**Q) Points A(–1, y) and B(5, 7) lie on a circle with centre O(2, –3y) such that AB is a diameter of the circle. Find the value of y. Also, find the radius of the circle.**

**Ans: **

**(i) value of y:**

We know that the center is the midpoint of the diameter,

Therefore if AB is the diameter and O is the centre of the circle,

then O will be midpoint of BD

and OA and OB will be equal and radii of the circle

Next, we know that the coordinates of midpoint are given by:

(X,Y) =

By substituting the given values, we get:

(2 , – 3y) =

By equating y coordinates, we get: – 3 y =

∴ – 3y x 2 = y + 7

∴ – 6 y = y + 7

∴ – 7 y = 7

∴ y = – 1

**Therefore, the value of y = – 1**

**(ii) Radius of circle:**

Radius of the circle is the length of line segment OA or OB

We have coordinates of Centre O (2, – 3 x -1) or (2, 3) and A (-1, -1)

We know that the distance between two points (X_{1}, Y_{1}) and (X_{2}, Y_{2}) is given by:

S = **√ **[(X_{2} – X_{1})^{2 } + (Y_{2} – Y_{1})^{2 }]

For length of line OA, we substitute the above coordinates values and get:

OA = **√ **[(- 1) – 2)^{2 } + (- 1 – 3)^{2 }]

= **√ **[(- 3)^{2 } + (- 4)^{2 }]

= **√ **( 9 + 16 )

= **√ **(25) = 5 units

**Therefore, the radius of the circle is 5 units.**

**Check:** Length of diameter = length of A(-1, -1) and B(5, 7)

*AB = *

*AB = = 10 units*

*Since diameter is 10 units, hence radius will be 5 units*

**Please press the “Heart”, if you liked the solution.**